The given quadratic function is:

$y = \frac{1}{2}x^2 + 3x - \frac{5}{2}$

We need to determine whether the function has a maximum or minimum value and find that value. Additionally, we need to find the value of $x$ at which this maximum or minimum occurs.

Step-by-Step Solution:

1. Identify the form of the quadratic function: A quadratic function is generally given by: $y = ax^2 + bx + c$ where $a = \frac{1}{2}$, $b = 3$, and $c = -\frac{5}{2}$.

2. Determine the direction of the parabola: The coefficient $a = \frac{1}{2}$ is positive, indicating that the parabola opens upwards. Therefore, the function has a minimum value.

3. Find the vertex of the parabola: The vertex form of a quadratic function gives the minimum (or maximum) point. The x-coordinate of the vertex is found using: $x = -\frac{b}{2a}$ Substituting $a$ and $b$: $x = -\frac{3}{2 \times \frac{1}{2}} = -\frac{3}{1} = -3$

4. Calculate the minimum value: Substitute $x = -3$ back into the original function to find $y$: $y = \frac{1}{2}(-3)^2 + 3(-3) - \frac{5}{2}$ $y = \frac{1}{2}(9) + (-9) - \frac{5}{2}$ $y = \frac{9}{2} - 9 - \frac{5}{2}$ $y = \frac{9}{2} - \frac{18}{2} - \frac{5}{2}$ $y = \frac{9 - 18 - 5}{2}$ $y = \frac{-14}{2}$ $y = -7$

Conclusion:

• The function $y = \frac{1}{2}x^2 + 3x - \frac{5}{2}$ has a minimum value.
• The minimum value is $-7$.
• This occurs when $x = -3$.

Completing the Given Statements:

函数y有最小 (选填1: 大; 2: 小) 值, 值 = -7. 当x = -3时，随x的增大而减小。

Would you like any further details or explanations?

Here are 5 questions you can ask next:

1. How do you find the vertex of a quadratic function in standard form?
2. Can you explain how to convert a quadratic function to vertex form?
3. What are the properties of parabolas in quadratic functions?
4. How do you determine the axis of symmetry for a quadratic function?
5. Can you explain the significance of the discriminant in quadratic equations?

Tip: When dealing with quadratic functions, always check the sign of the leading coefficient to determine if the parabola opens upwards (minimum value) or downwards (maximum value).